Continuity at a point

Let $f:(X,d_X)\to(Y,d_Y)$ and let $a\in X$.
We say $f$ is continuous at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for all $x\in X$,

In words: points $x$ sufficiently close to $a$ must have images $f(x)$ close to $f(a)$.

Sequential characterization (metric spaces): $f$ is continuous at $a$ iff whenever $x_n\to a$, we have $f(x_n)\to f(a)$.
This reframes continuity using limits of sequences .

Examples:

• $f(x)=x^2$ is continuous at every $a\in\mathbb{R}$.
• The step function $f(x)=\mathbf{1}_{(0,\infty)}(x)$ is not continuous at $0$.

If $f$ is continuous at every point of a set $A$, then $f$ is continuous on \(A\) .